I want to understand the adjunction $\mathbf{Ind}^G_H \dashv \mathbf{Res}^G_H $ between the categories of representations of a group $G$ and of its subgroup $H$.
In particular, I want to understand how to calculate the equivariant maps in the following case. Please tell me what I am miscalculating, misinterpreting or misunderstanding!
Consider the permutation group $G=S_3=\{(),(12),(13),(23),(12)(13)=(123),(13)(12)=(132)\}$ with subgroup $H=S_2=\{(),(12)\}$. Define vector spaces $U=\mathbb{C}^2$ and $V=\mathbb{C}^3$.
Let $\theta_U$ be the representation of $H$ that maps $()\rightarrow \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, $(12) \rightarrow \begin{pmatrix} -1 & 1 \\ 0 & 1 \end{pmatrix} $.
Let $\phi_V$ be the permutation representation of $G$ given by $()\rightarrow \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, (12) \rightarrow \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, (13) \rightarrow \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}, (23) \rightarrow \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, (123) \rightarrow \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, (132) \rightarrow \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$
Then any equivariant map $\alpha=(a_{ij}):(\theta_U,U)\rightarrow (\mathbf{Res}^G_H(\phi_V),V)$ satisfies a trivial equation for the identity element $()$ and also satisfies the following equation for $(12)$.
$ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix} \begin{pmatrix} -1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} $
The solution $\alpha = \begin{pmatrix} a_{11} & a_{12} \\ -a_{11} & a_{11}+a_{12} \\ 0 & a_{32} \end{pmatrix}$ characterizes the equivariant maps in $\mathbf{Rep}_H((U,\theta_U), \mathbf{Res}^G_H(V,\phi_V))$ as a family dependent on the three parameters $a_{11}$, $a_{12}$, $a_{32}$.
But I do not get such a family when I try to calculate the equivariant maps in $\mathbf{Rep}_G(\mathbf{Ind}^G_H(U,\theta_U),(V,\phi_V))$. What am I miscalculating or misunderstanding?
First, I consider how the elements of $G$ act on left coset representatives $()$, $(13)$, $(23)$ of $G/H$.
$ \begin{matrix} \times & \mathbf{()} & \mathbf{(13)} & \mathbf{(23)} \\ \mathbf{()} & () & (13) & (23) \\ \mathbf{(12)} & (12) & (23)(12) & (13)(12)\\ \mathbf{(13)} & (13) & () & (23)(12) \\ \mathbf{(13)(12)} & (13)(12) & (23) & (12) \\ \mathbf{(23)} & (23) & (13)(12) & () \\ \mathbf{(23)(12)} & (23)(12) & (12) & (13) \\ \end{matrix} $
Then I express these actions with permutation matrices of blocks, as follows, where I express the action of the subgroup representation by $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ for $()$ and by $\begin{pmatrix} -1 & 1 \\ 0 & 1 \end{pmatrix}$ for $(12)$. I have omitted the blocks of zeroes:
$ () \rightarrow \begin{pmatrix} 1 & 0 & & & & \\ 0 & 1 & & & & \\ & & 1 & 0 & & \\ & & 0 & 1 & & \\ & & & & 1 & 0 \\ & & & & 0 & 1 \end{pmatrix}, (12) \rightarrow \begin{pmatrix} -1 & 1 & & & & \\ 0 & 1 & & & & \\ & & & & -1 & 1 \\ & & & & 0 & 1 \\ & & -1 & 1 & & \\ & & 0 & 1 & & \end{pmatrix}$, $(13) \rightarrow \begin{pmatrix} & & 1 & 0 & & \\ & & 0 & 1 & & \\ 1 & 0 & & & & \\ 0 & 1 & & & & \\ & & & & -1 & 1 \\ & & & & 0 & 1 \end{pmatrix}, (13)(12) \rightarrow \begin{pmatrix} & & -1 & 1 & & \\ & & 0 & 1 & & \\ & & & & 1 & 0 \\ & & & & 0 & 1 \\ -1 & 1 & & & & \\ 0 & 1 & & & & \end{pmatrix}$ $ (23) \rightarrow \begin{pmatrix} & & & & 1 & 0 \\ & & & & 0 & 1 \\ & & -1 & 1 & & \\ & & 0 & 1 & & \\ 1 & 0 & & & & \\ 0 & 1 & & & & \end{pmatrix}, (23)(12) \rightarrow \begin{pmatrix} & & & & -1 & 1 \\ & & & & 0 & 1 \\ -1 & 1 & & & & \\ 0 & 1 & & & & \\ & & 1 & 0 & & \\ & & 0 & 1 & & \end{pmatrix}$
Then I get six equations (one for each element of $G$) of the form
$ \begin{pmatrix} c_{11} & c_{12} & c_{13} & c_{14} & c_{15} & c_{16} \\ c_{21} & c_{22} & c_{23} & c_{24} & c_{25} & c_{26} \\ c_{31} & c_{32} & c_{33} & c_{34} & c_{35} & c_{36} \end{pmatrix} \begin{pmatrix} -1 & 1 & & & & \\ 0 & 1 & & & & \\ & & & & -1 & 1 \\ & & & & 0 & 1 \\ & & -1 & 1 & & \\ & & 0 & 1 & & \end{pmatrix} \begin{pmatrix} u_{11} \\ u_{12} \\ u_{21} \\ u_{22} \\ u_{31} \\ u_{32} \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} c_{11} & c_{12} & c_{13} & c_{14} & c_{15} & c_{16} \\ c_{21} & c_{22} & c_{23} & c_{24} & c_{25} & c_{26} \\ c_{31} & c_{32} & c_{33} & c_{34} & c_{35} & c_{36} \end{pmatrix} \begin{pmatrix} u_{11} \\ u_{12} \\ u_{21} \\ u_{22} \\ u_{31} \\ u_{32} \end{pmatrix} $
I am working on that here.
When I solve them, I get the equivariant map
$ \gamma = C \begin{pmatrix} 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 \end{pmatrix} $
which I don't think makes sense. I'm expecting to see something related to the equivariant map that I got for the restricted representation. I imagine that there should be three parameters. Please, what am I doing wrong?